Question

a)

Let A and B be sets s.t. the cartesian product is defined as AB={(a,b)aA,bB}.

if G and H are groups, show that GH forms a group under the operation defined as

(g1,h1)(g2,h2)=(g1g2,h1h2) .

b)

Prove that the direct product GH of two groups has a subgroup isomorphic to G and a subgroup isomorphic to H

Answer

a)

Let g1,g2G and h1,h2H . Since closure for G and H are given as they are groups, then (g1,h1)(g2,h2)=(g1g2,h1h2)GH . Thus closure holds for GH holds.

For associativity; Let (g1,h1),(g2,h2) and (g3,h3)GH , then

((g1,h1)(g2,h2))(g3,h3)=(g1g2h1h2)(g3,h3)=((g1g2)g3,(h1h2)h3)=(g1(g2g3),h1(h2h3))=(g1,h1)((g2,h2)(g3,h3)) .

Thus GH is associative.

For the identity

Let (g,h)GH , then

(g,h)(eG,eH)=((eGg),(eHh))=(g,h)=(eG,eH)(g,h)=((eHh),(eGg))=(eG,eH)(g,h) .

Thus identity exists for GH .

For the inverse

Since G and H are groups; g1Gs.t.gg1=eG=g1g,h1Hs.t.hh1=eH=h1h,where eG is the identity for G and eH is the identity for H .

Therefore (g1,h1)(g,h)=(g1g,h1h)=(eG,eH)=(gg1,hh1)=(g,h)(g1,h1) .

Thus the identity exists for GH .

It follows that GH is a group under the operation .

b)